Theorem 1.2. If all non-identity elements of G have the same order, this order is a prime p and |G| is a power of p. Proof. If |G| has two prime factors, say p and q, then G contains elements of orders p and q by Cauchy, which contradicts the hypothesis. Thus |G| has only one prime factor, say |G| = pm for a prime p. The orders of a non-identity element could be {p, p2, . . . , pm}. However, by...

A theorem now known as Sperner's Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal [n/2\. (We denote by |*J, resp. [*], the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest anti...

Partial orders have applications in various areas in computer science such as distributed systems, object oriented languages, knowledge representation systems and databases. In this paper we present a new technique to encode partially ordered sets or posets. Specifically, we use the family of two-dimensional posets, which we refer to as rectangles, to encode a given poset; an element x is less ...

In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group PSL(13, 2) using only the set of its element orders.

We consider the problem of searching for a given element in a partially ordered set. More precisely, we address the problem of computing efficiently near-optimal search strategies for typical partial orders. We consider two classical models for random partial orders, the random graph model and the uniform model. We shall show that certain simple, fast algorithms are able to produce nearly-optim...

A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo’s), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain a...

A theorem now known as Sperner’s Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal bn=2c. (We denote by bxc, resp. dxe, the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest anti...

in the present study, we reported the results of the monitoring and investigations of the ichthyofauna of the beshar river, one of the tributaries of the tigris−euphrates basin in kohkiluyeh and boyer-ahmad province, southwest of iran. the results showed that during a period from 2010 to 2011, a total of 13 fish species belonging to 4 families and 3 orders were recorded in the beshar river. the...